Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity
Adel M. Al‐Mahdi
Abstract
Abstract In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>≤</mml:mo><mml:mo>−</mml:mo><mml:mi>ξ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace/><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math> where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.